Section 3: Hydraulic Operation of Culverts

Parameters

The hydraulic operation and performance of a culvert involve
a number of factors. You must determine, estimate, or calculate
each factor as part of the hydraulic design or analysis.

The following procedures assume steady flow but can involve
extensive calculations that lend themselves to computer application.
The procedures supersede simplified hand methods of other manuals.
TxDOT recommends computer models for all final design applications,
although hand methods and nomographs may be used for initial planning.

Headwater under Inlet Control

Inlet control occurs when the culvert barrel is capable of
conveying more flow than the inlet will accept. Inlet control is
possible when the culvert slope is hydraulically steep (dc >
du). The control section of a culvert operating
under inlet control is located just inside the entrance. When the
flow in the barrel is free surface flow, critical depth occurs at
or near this location, and the flow regime immediately downstream
is supercritical. Depending on conditions downstream of the culvert
inlet, a hydraulic jump may occur in the culvert. Under inlet control,
hydraulic characteristics downstream of the inlet control section
do not affect the culvert capacity. Upstream water surface elevation
and inlet geometry (barrel shape, cross-sectional area, and inlet
edge) are the major flow controls.

A fifth-degree polynomial equation based on regression analysis
is used to model the inlet control headwater for a given flow. Analytical
equations based on minimum energy principles are matched to the
regression equations to model flows that create inlet control heads
outside of the regression data range. Equation 8-4 only applies
when 0.5 ≤ HWic/D ≤ 3.0.

Determine the
potential head from the centroid of the culvert opening, which is
approximated as the sum of the invert elevation and one half the
rise of the culvert. The effective area, A, and orifice coefficient,
C, are implicit.

Generally for TxDOT designs, it is not considered efficient
to design culverts for HWi/D < 0.5. However,
if such a condition is likely, an open channel flow minimum energy
equation (weir equation) should be used, with the addition of a
velocity head loss coefficient. The minimum energy equation, with
the velocity head loss adjusted by an entrance loss coefficient,
generally describes the low flow portion of the inlet control headwater
curve. However, numerical errors in the calculation of flow for
very small depths tend to increase the velocity head as the flow
approaches zero. This presents little or no problem in most single
system cases because the flows that cause this are relatively small.

In many of the required calculations for the solution of multiple
culverts, the inlet control curve must decrease continuously to
zero for the iterative calculations to converge. Therefore, computer models
modify this equation to force the velocity head to continually decrease
to zero as the flow approaches zero.

The “Charts” in
HDS-5 (FHWA,
Hydraulic Design of Highway Culverts) provide guidance for graphical
solution of headwater under inlet control.

Headwater under Outlet Control

Outlet control occurs when the culvert barrel is not capable
of conveying as much flow as the inlet opening will accept. Outlet
control is likely only when the hydraulic grade line inside the
culvert at the entrance exceeds critical depth. (See Chapter 6 for
Hydraulic Grade Line Analysis.) Therefore, outlet control is most
likely when the culvert is on a mild slope (dn >
dc). It is also possible to experience outlet
control with a culvert on a steep slope (dn <
dc ) with a high tailwater such that subcritical
flow or full flow exists in the culvert.

The headwater of a culvert in outlet control is a function
of discharge, conduit section geometry, conduit roughness characteristics,
length of the conduit, profile of the conduit, entrance geometry (to
a minor extent), and (possibly) tailwater level.

The headwater of a culvert under outlet control can be adjusted,
for practical purposes, by modifying culvert size, shape, and roughness.
Both inlet control and outlet control need to be considered to determine
the headwater. The following table provides a summary conditions
likely to control the culvert headwater. Refer to Figure 8‑4 and
Figure 8‑5 to identify the appropriate procedures to make the determination.

Outlet control headwater is determined by accounting for the
total energy losses that occur from the culvert outlet to the culvert
inlet. Figure 8‑7 and Figure 8‑8 and associated procedures in Section
4 should be used to analyze or design a culvert.

Outlet control headwater HWoc depth
(measured from the flowline of the entrance) is expressed in terms
of balancing energy between the culvert exit and the culvert entrance
as indicated by Equation 8-8.

For convenience energy balance at outlet, energy losses through
barrel, and energy balance at inlet should be considered when determining
outlet control headwater.

When the tailwater controls the outlet flow, Equation 8-10
represents the energy balance equation at the conduit outlet. Traditional
practice has been to ignore exit losses. If exit losses are ignored, the
hydraulic grade line inside the conduit at the outlet,outlet
depth, Ho, should be assumed to be the same
as the hydraulic grade line outside the conduit at the outlet and
Equation 8-10 should not be used.

Tailwater depth (TW) is lower
than critical depth (dc) in culvert at outlet

Slope is hydraulically mild

Set Ho as
critical depth.

Uniform depth is higher than
top of the barrel

Slope is hydraulically steep

Set Ho as
the higher of the barrel depth (D) and depth using Equation 8-10.

Uniform depth is lower than
top of barrel and tailwater exceeds critical depth

Slope is hydraulically steep

Set Ho using
Equation 8-10.

Uniform depth is lower than
top of barrel and tailwater is below critical depth

Slope is hydraulically steep

Ignore, as outlet control is
not likely.

NOTE: For hand computations and some computer programs, Ho is
assumed to be equal to the tailwater depth (TW). In such a case,
computation of an exit head loss (ho) would
be meaningless since the energy grade line in the culvert at the
outlet would always be the sum of the tailwater depth and the velocity
head inside the culvert at the outlet (hvo).

Free Surface Flow (Type A)

If free surface flow is occurring in the culvert, the hydraulic
parameters are changing with flow depth along the length of the
culvert as seen in Figure 8‑9. It is necessary to calculate the
backwater profile based on the outlet depth, Ho.

By definition, a free-surface backwater from the outlet end
of a culvert may only affect the headwater when subcritical flow
conditions exist in the culvert. Subcritical, free-surface flow
at the outlet will exist if the culvert is on a mild slope with
an outlet depth (Ho) lower than the outlet
soffit or if the culvert is on a steep slope with a tailwater higher
than critical depth at the culvert outlet and lower than the outlet
soffit.

The
Direct
Step Backwater Method is used to determine the water surface
profile (and energy losses) though the conduit. The depth, Ho, is
used as the starting depth, d1. For subcritical flow, the calculations
begin at the outlet and proceed in an upstream direction. Use the
depth, Ho, as the starting depth, d1,
in the Direct Step calculations.

When using the direct step method, if the inlet end of the
conduit is reached without the calculated depth exceeding the barrel
depth (D), it verifies that the entire length of the conduit is
undergoing free surface flow. Set the calculated depth (d2)
at the inlet as Hi and refer to Energy Balance
at Inlet to determine the headwater.

When using the direct step method, if the calculated depth
(d2) reaches or exceeds the barrel depth (D),
the inside of the inlet is submerged. Refer to Type AB - Free surface
at outlet and full flow at inlet for a description. This condition
is possible if the theoretical value of uniform depth is higher than
the barrel depth.

Full Flow in Conduit (Type B)

If full flow is occurring in the conduit, rate of energy losses
through the barrel is constant (for steady flow) as seen in Figure
8‑10. The hydraulic grade line is calculated based on outlet depth, Ho,
at the outlet.

Full flow at the outlet occurs when the outlet depth (Ho)
equals or exceeds barrel depth D. Full flow is maintained throughout
the conduit if friction slope is steeper than conduit slope, or
if friction slope is flatter than conduit slope but conduit is not
long enough for the hydraulic grade line to get lower than the top
of the barrel.

Equation 8-13 is used to calculate friction slope. If friction
slope is flatter than the conduit slope, the hydraulic grade line
may drop below the top of the barrel. If this occurs, refer to Type
BA - Full Flow at the outlet and free surface flow at the inlet.

Full Flow at Outlet and Free Surface Flow at Inlet (Type BA)

If the friction slope is flatter than the conduit slope, it
is possible that full flow may not occur along the entire length
of the culvert (see the Table 8-5 on
Entrance
Loss Coefficients). The following steps should be followed:

Determine Type BA free surface losses,
if applicable. Free surface flow begins at the point of intersection
of the hydraulic grade line and the soffit of the culvert barrel
as shown in Figure 8‑11. If this condition occurs, determine the
depth of flow at the inlet using the Direct Step Method with the
starting depth (d1) equal to the barrel rise
(D) and starting at the location along the barrel at which free
surface flow begins.

Determine Type BA hydraulic grade line
at inlet, if applicable. When the inlet end of the conduit is reached
using the direct step method, set the calculated depth at the inlet
as Hi and refer to Energy Balance at Inlet
to determine the headwater.

Free Surface at Outlet and Full Flow at Inlet (Type AB)

When the outlet is not submerged, full flow will begin within
the conduit if the culvert is long enough and the flow high enough.
Figure 8‑12 illustrates this condition. This condition is possible if
the theoretical value of uniform depth is higher than the barrel
depth. The following steps should be followed:

Determine Type AB free surface losses,
if applicable. Refer to Water Surface Profile Calculations, Free
Surface Flow to determine the water surface profile in the conduit.
If the computed depth of flow reaches or exceeds the barrel depth
before the end of the conduit, note the position along the conduit
at which this occurs and proceed to full flow losses below. Otherwise, complete
the procedure described under Free Surface Flow.

Determine Type AB full flow losses, if
applicable. Begin full flow calculations at the point along the
conduit where the computed water surface intersects the soffit of
the barrel as determined above. Determine the energy losses through
the remainder of the conduit using Equation 8-11 but substituting
Lf, the remaining conduit length, for L.

Determine Type AB hydraulic grade line
at inlet, if applicable. Compute the depth of the hydraulic grade
line, Hi, at the inside of the inlet end
of the conduit using Equation 8-12. Use the barrel height D as the
starting hydraulic grade line depth in place of Ho, and
use the remaining length, Lf, in place of
L. Refer to Energy Balance at Inlet to determine headwater depth.

Energy Balance at Inlet

The outlet control headwater, HWoc,
is computed by balancing the energy equation, depicted as Equation
8-15. The hydraulic grade at the inside face of the culvert at the
entrance will need to be known. See
Energy
Losses through Conduit. The velocity at the entrance (vi)
is used to compute the velocity head at the entrance (hvi).

Hi=
depth of hydraulic grade line just inside the culvert at inlet (ft.
or m).

Generally, when using Equation 8-15, the velocity approaching
the entrance may be assumed to be negligible so that the headwater
and energy grade line are coincident just upstream of the upstream face
of the culvert. This is conservative for most department needs.
The approach velocity may need to be considered when performing
the following tasks:

NOTE: The pipes of pipe runner SETs have been proven to be
within the tolerance of the entrance loss equations. Therefore,
the entrance should be evaluated solely for its shape and the effect
of the pipes should be ignored.

If the conduit will flow full at the outlet,
usually due to a high tailwater or a conduit capacity lower than
the discharge, set do to the barrel rise
(D) so that the full cross-sectional area of the conduit is used
as shown in Figure 8‑14.

Depth Estimation Approaches

Use a step backwater
method starting from critical depth (dc)
at the inlet and proceed downstream to the outlet: If the tailwater
is lower than critical depth at the outlet, calculate the velocity
resulting from the computed depth at the outlet. If the tailwater
is higher than critical depth, a hydraulic jump within the culvert
is possible. The
Hydraulic
Jump in Culverts subsection below discusses a means of estimating
whether the hydraulic jump occurs within the culvert. If the hydraulic
jump does occur within the culvert, determine the outlet velocity
based on the outlet depth, do = Ho.

Assume uniform depth at the outlet. If
the culvert is long enough and tailwater is lower than uniform depth,
uniform depth will be reached at the outlet of a steep slope culvert:
For a short, steep culvert with tailwater lower than uniform depth,
the actual depth will be higher than uniform depth but lower than
critical depth. This assumption will be conservative; the estimate
of velocity will be somewhat higher than the actual velocity. If
the tailwater is higher than critical depth, a hydraulic jump is
possible and the outlet velocity could be significantly lower than
the velocity at uniform depth.

Direct Step Backwater Method

The Direct Step Backwater Method uses the same basic equations
as the Standard Step Backwater Method but is simpler to use because
no iteration is necessary. In the Direct Step Method, an increment
(or decrement) of water depth (δd) is chosen and
the distance over which the depth change occurs is computed. The
accuracy depends on the size of δd. The method
is appropriate for prismatic channel sections such as occur in most
conduits. It is useful for estimating supercritical profiles and
subcritical profiles.

Choose a starting
point and starting water depth (d1). This
starting depth depends on whether the profile is supercritical or
subcritical. Generally, for culverts, refer to outlet depth and
set d1 to the value of H0.
Otherwise, you may use the following conditions to establish d1:

For a mild slope
(dc < du) and free
surface flow at the outlet, begin at the outlet end. Select the
higher of critical depth (dc) and tailwater
depth (TW). Supercritical flow may occur in a culvert on a mild
slope. However, most often, the flow will be subcritical when mild slopes
exist. Check this assumption.

For a steep slope in which tailwater depth
is lower than critical depth, begin the water surface profile computations
at the culvert entrance starting at critical depth and proceed downstream
to the culvert exit. This implies inlet control, in which case the
computation may be necessary to determine outlet velocity but not
headwater.

For a submerged outlet in which free surface
flow begins along the barrel, use the barrel depth, D, as the starting
depth. Begin the backwater computations at the location where the
hydraulic grade line is coincident with the soffit of the culvert.

The following steps assume subcritical
flow on a mild slope culvert for a given discharge, Q, through a
given culvert of length, L, at a slope, So.
Calculate the following at the outlet end of the culvert based on
the selected starting depth (d1):

Choose an increment or decrement of flow
depth, d: if d1 >
du, use a decrement (negative d); otherwise, use an increment.The
increment, d, should
be such that the change in adjacent velocities is not more than
10%.

Consider the new depth and location to
be the new starting positions (assign the subscript 1 to those
values currently identified with the subscript 2)
and repeat steps 3 to 7, summing the incremental lengths, L, until the total length, L, equals or just exceeds the length of the culvert.
You may use the same increment throughout or modify the increment
to achieve the desired resolution. Such modifications are necessary
when the last total length computed far exceeds the culvert length
and when high friction slopes are encountered. If the computed depth
reaches the barrel rise (D) before reaching the culvert inlet, skip
step 9 and refer to the
Type
AB full flow losses to complete the analysis.

Subcritical Flow and Steep Slope

If the depth, d, reaches critical depth
before the inlet of the culvert is reached, the headwater is under
inlet control (
Headwater
Under Inlet Control) and a
hydraulic
jump may occur in the culvert barrel (refer to the following
subsection for discussion of the hydraulic jump in culverts).

If the depth at the inlet is higher than
critical depth, determine the outlet control head using Equation
8-15 as discussed in the
Energy
Balance at Inlet subsection. A hydraulic jump may occur within
the culvert.

Hydraulic Jump in Culverts

Figure 8-15 provides a sample plot of depth and momentum function
and an associated specific energy plot. For a given discharge and
given energy and momentum, there exist two possible depths, one
less than critical depth (supercritical flow) a sequent (or conjugate)
depth greater than critical depth (subcritical flow). With a proper
configuration, the water flowing at the lower depth in supercritical
flow can “jump” abruptly to its sequent depth in subcritical flow.
This is called a hydraulic jump. With the abrupt change in flow
depth comes a corresponding change in cross-sectional area of flow
and a resulting decrease in average velocity.

By comparing the two curves at a supercritical depth and its
sequent depth, it is apparent that the hydraulic jump involves a
loss of energy. Also, the momentum function defines critical depth
as the point at which minimum momentum is established.

The term A represents
the first moment of area about the water surface. Assuming no drag
forces or frictional forces at the jump, conservation of momentum
maintains that the momentum function at the approach depth, M1,
is equal to the momentum function at the sequent depth, Ms.

The potential occurrence of the hydraulic jump within the
culvert is determined by comparing the outfall conditions with the
sequent depth of the supercritical flow depth in the culvert. The
conditions under which the hydraulic jump is likely to occur depend
on the slope of the conduit.

Under mild slope conditions (dc <
du) with supercritical flow in the upstream
part of the culvert, the following two typical conditions could
result in a hydraulic jump:

Sequent Depth

A direct solution for sequent depth, ds is
possible for free surface flow in a rectangular conduit on a flat
slope using Equation 8-23. If the slope is greater than about 10
percent, a more complex solution is required to account for the
weight component of the water. FHWA
Hydraulic
Engineering Circular 14 provides more detail for such conditions.

Equation 8-24 applies to other conduit shapes having slopes
of about 10 percent or less. The first moment of area about the
surface, A, is
dependent on the shape of the conduit and depth of flow. A relationship
between flow depth and first moment of area must be acquired or
derived.

Roadway Overtopping

Where water flows both over the roadway and through a culvert
(see Figure 8‑17), a flow distribution analysis is required to define
the hydraulic characteristics. This is a common occurrence where a
discharge of low design AEP (low probability of occurrence) is applied
to a facility designed for a lower design frequency.

For example, a complete design involves the application and
analysis of a 1% AEP discharge to a hydraulic facility designed
for a much smaller flood. In such a case, the headwater may exceed
the low elevation of the roadway, causing part of the water to flow
over the roadway embankment while the remainder flows through the
structure. The headwater components of flow form a common headwater
level. An iterative process is used to establish this common headwater.

The following procedure is an iterative approach that is reasonable
for hand computations and computer programs:

Initially assume
that all the runoff (analysis discharge) passes through the culvert,
and determine the headwater. Use the procedures outlined in the
Culvert
Design section. If the headwater is lower than the low roadway
elevation, no roadway overtopping occurs and the analysis is complete.
Otherwise, proceed to step 2.

Record the analysis discharge as the initial
upper flow limit and zero as the initial lower flow limit. Assign
50% of the analysis discharge to the culvert and the remaining 50%
to the roadway as the initial apportionment of flow.

Base the value Hh on
the assumption that the effective approach velocity is negligible.
For estimation of maximum headwater, this is a conservative assumption.
However, under some conditions, such as the need to provide adequate
detention storage, you may need to consider the approach velocity
head (v2/2g). That is, replace Hh in
Equation 8-28 with Hh + v2/2g.

With reference to Figure 8‑19, the flow
over the embankment will not be affected by tailwater if the excess
(Ht) is lower than critical depth of flow
over the road (approximately 0.67 Hh) . For
practical purposes, Ht/Hh may
approach 0.8 without any correction coefficient. For Ht/Hh values
above 0.8 use Figure 8‑18 to determine kt.

For most cases of flow over highway embankments,
the section over which the discharge must flow is parabolic or otherwise
irregular (see Figure 8‑20). In such cases, it becomes necessary
to divide the section into manageable increments and to calculate
individual weir flows for the incremental units, summing them for
total flow.

If the tailwater is sufficiently high,
it may affect the flow over the embankment. In fact, at high depth,
the flow over the road may become open channel flow, and weir calculations are
no longer valid. At extremely high depth of roadway overtopping,
it may be reasonable to ignore the culvert opening and compute the
water surface elevation based on open channel flow over the road.

Add the calculated roadway overflow to
the culvert flow. If the calculated total is greater than the analysis
discharge, record the current culvert flow apportionment as the
current upper flow limit and set the new culvert flow apportionment
at a value halfway between the current upper and lower flow limits.
If the calculated total is less than the analysis discharge, record
the current culvert flow apportionment as the lower flow limit for
the culvert and set the new culvert flow apportionment at a value
halfway between the current upper and lower flow limits.

Repeat steps 3 to 5, using the culvert
flow apportionment established in step 5, until the difference between
the current headwater and the previous headwater is less than a
reasonable tolerance. For computer programs, the department recommends
a tolerance of about 0.1 in. The current headwater and current assigned
culvert flow and calculated roadway overflow can then be considered
as the final values.

Performance Curves

For any given culvert, the control (outlet or inlet) might
vary with the discharge. Figure 8‑18 shows sample plots of headwater
versus discharge for inlet and outlet control. The envelope (shown
as the bold line) represents the highest value of inlet and outlet
headwater for any discharge in the range. This envelope is termed
a performance curve.

In this example, inlet control prevails at lower discharges
and flow transitions to outlet control as the discharge increases.
The flatter portion represents the effect of roadway overflow. The
performance curve can be generated by calculating the culvert headwater
for increasing values of discharge. Such information is particularly
useful for performing risk assessments and for hydrograph routing
through detention ponds and reservoirs.

Exit Loss Considerations

The traditional assumption in the design of typical highway
culverts is continuity of the hydraulic grade line. At the outlet,
this implies that when the tailwater is higher than critical depth
and subcritical flow exists, the hydraulic grade line immediately
inside the barrel is equal to the tailwater level. This is reasonable
for most normal culvert designs for TxDOT application. However,
by inference there can be no accommodation of exit losses because
the energy grade line immediately inside the culvert can only be
the hydraulic grade line plus the velocity head, no matter what
the velocity is in the outfall.

Occasionally, an explicit exit loss may need to be accommodated.
Some examples are as follows:

design of detention pond control structures
in which storage volumes are sensitive to small changes in elevation.

If such a need arises, base the starting hydraulic grade level
(Ho) is based on balancing
Equation
8-29 between the outside and inside of the culvert face at
the outlet. A common expression for exit loss appears in Equation
8-30. This assumes that the tailwater velocity (vTW)
is lower than the culvert outlet velocity (vo)
and the tailwater is open to the atmosphere. If the above approach
is used, it is most likely that the outlet depth (Ho)
will be lower than the tailwater. This conforms to basic one-dimensional
hydrostatic principles.